NECTA BASIC MATHEMATICS 2025 SOLUTION

NECTA Form 4 Basic Mathematics 2025

Suggested Solutions & Step-by-Step Calculations

SECTION A (60 Marks)

1. Numbers and Approximation

(a) Express 0.1136 as a simple fraction:
0.1136 = 1136 / 10000
Divide both numerator and denominator by 16:
1136 ÷ 16 = 71
10000 ÷ 16 = 625
Ans: 71 / 625

(b) Rounding off:
(i) 0.00070482 to 3 significant figures: First non-zero is 7. Count three digits: 7, 0, 4. The next is 8, so round up.
Ans: 0.000705
(ii) 1,233,388 to nearest ten thousands: Look at the thousands digit (3). It is less than 5, so round down.
Ans: 1,230,000

(c) Percentage:
Pupils from city = 3/5.
Pupils NOT from city = 1 - 3/5 = 2/5.
Percentage = (2/5) × 100% = 40%.
Ans: 40%

2. Exponents, Surds, and Logarithms

(a) Exponents:
Expression: 27ⁿ × 9²ⁿ × 3
Convert to base 3:
= (3³)ⁿ × (3²)²ⁿ × 3¹
= 3³ⁿ × 3⁴ⁿ × 3¹
Add powers: 3n + 4n + 1
Ans: 3^{7n + 1}

(b) Rationalize the denominator:
Expression: (√2 + 1) / (√2 - √5)
Multiply by conjugate (√2 + √5) / (√2 + √5):
Numerator: (√2 + 1)(√2 + √5) = 2 + √10 + √2 + √5
Denominator: (√2)² - (√5)² = 2 - 5 = -3
Ans: -(2 + √10 + √2 + √5) / 3

(c) Logarithms:
Find log 72 given log 2 = 0.3010, log 3 = 0.4771.
72 = 8 × 9 = 2³ × 3²
log 72 = log(2³) + log(3²)
= 3 log 2 + 2 log 3
= 3(0.3010) + 2(0.4771)
= 0.9030 + 0.9542
Ans: 1.8572

3. Sets and Probability

(a) Sets:
Total (U) = 260, Physics (P) = 130, Chem (C) = 150, Both (P ∩ C) = 40.
(i) Physics only = n(P) - n(P ∩ C) = 130 - 40 = 90.
(ii) Neither = n(U) - n(P ∪ C)
n(P ∪ C) = n(P) + n(C) - n(P ∩ C) = 130 + 150 - 40 = 240.
Neither = 260 - 240 = 20.

(b) Probability:
Total = 35. Goats (G) = 18, Cows (C) = 20, Both = 3.
Goats only = 18 - 3 = 15.
Probability = (Goats only) / Total = 15 / 35.
Simplify by dividing by 5:
Ans: 3/7

4. Geometry and Perimeter

(a) Area of Triangle:
Angle ABC = 180° - 150° = 30° (Linear pair).
Area = 0.5 × AB × BC × sin(B)
22.6 = 0.5 × 8 × BC × sin(30°)
22.6 = 4 × BC × 0.5
22.6 = 2 × BC → BC = 11.3 cm.
Ans: BC = 11.3 cm

(b) Regular Pentagon Inscribed in Circle:
Radius r = 10m. Pentagon has 5 sides.
Angle at center = 360 / 5 = 72°.
Using sine rule for side length s: s = 2r sin(72/2) = 2(10) sin(36°).
s = 20 × 0.5878 = 11.756 m.
Perimeter = 5 × s = 5 × 11.756
Ans: 58.78 m

5. Rates and Variations

(a) Currency Conversion:
Rate: 2300 TZS = 1 USD.
Amount: 27,600,000 TZS.
USD = 27,600,000 / 2,300
Ans: 12,000 USD

(b) Joint Variation:
h ∝ V / r² → h = k V / r².
Given h=3, V=900, r=10:
3 = k(900) / 100 → 3 = 9k → k = 1/3.
Find h when V=1200, r=5:
h = (1/3) × 1200 / 5²
h = 400 / 25
Ans: h = 16 cm

6. Vectors and Coordinate Geometry

(a) Vectors:
a = 2i - j, b = -i + 3j, c = 3i - 4j.
Sum = (2 - 1 + 3)i + (-1 + 3 - 4)j
Sum = 4i - 2j.
Magnitude = √(4² + (-2)²) = √(16 + 4) = √20.
Ans: 2√5

(b) Coordinate Midpoint:
Midpoint M(3, 5). Point A(6, 7). Let Point B be (x, y).
(6 + x)/2 = 3 → 6 + x = 6 → x = 0.
(7 + y)/2 = 5 → 7 + y = 10 → y = 3.
Ans: (0, 3)

7. Ratios and Accounts

(a) Concrete Mixture:
Ratio Gravel:Sand:Cement = 500:600:200 = 5:6:2.
Total parts = 5 + 6 + 2 = 13.
Total Mass = 39,000 kg.
Cement = (2 / 13) × 39,000 = 2 × 3,000
Ans: 6,000 kg

(b) Net Profit:
Cost of Goods Sold (COGS) = Opening Stock + Purchases - Closing Stock
COGS = 2,000 + 13,000 - 500 = 14,500.
Gross Profit = Sales - COGS = 30,000 - 14,500 = 15,500.
Net Profit = Gross Profit - Expenses = 15,500 - 6,500.
Ans: 9,000/=

8. Compound Interest and Progression

(a) Compound Interest:
Formula: A = P(1 + R/100)ⁿ
A = 10,000(1 + 5/100)³
A = 10,000(1.05)³ = 10,000(1.157625)
Ans: 11,576.25 TZS

(b) Geometric Progression (GP):
Terms: 3, x-1, 27.
Property of GP: middle term squared = product of outer terms.
(x - 1)² = 3 × 27 = 81.
x - 1 = ±9.
x = 9 + 1 = 10 OR x = -9 + 1 = -8.
Ans: x = 10 or x = -8

9. Trigonometry

(a) Largest Angle (Cosine Rule):
Sides 6, 7, 8. Largest angle (A) is opposite side 8.
8² = 6² + 7² - 2(6)(7)cosA
64 = 36 + 49 - 84cosA
64 = 85 - 84cosA
-21 = -84cosA → cosA = 0.25
A = cos^-1(0.25)
Ans: ≈ 75.5°

(b) Right Angled Triangle:
Right angle at B. CA = 6 (Hypotenuse), AB = 3 (Adjacent to A).
cos A = 3/6 = 0.5 → A = 60°.
Therefore, C = 180 - 90 - 60 = 30°.
tan C = tan 30° = 1/√3.
Length BC (Pythagoras): BC = √(6² - 3²) = √27.
Ans: BC = 3√3 cm (or 5.2 cm), tan C = 1/√3

10. Algebra

(a) Quadratic Form:
Given 2/t¹⁰ - 3/t⁵ + 1 = 0.
Let x = 1/t⁵. Then x² = 1/t¹⁰.
Substitute: 2x² - 3x + 1 = 0.
Ans: 2x² - 3x + 1 = 0

(b) Solve for x:
Factorize: (2x - 1)(x - 1) = 0.
x = 1/2 or x = 1.
Ans: x = 0.5 or x = 1

SECTION B (40 Marks)

11. Statistics

(a) Frequency:
Reading from the table, for class 75-79.
Ans: 55 students

(b) Median:
Total N = 100. Median position = N/2 = 50.
Cumulative frequency before 75-79 class = 10 (65-69) + 12 (70-74) = 22.
Median Class is 75-79. Lower limit L = 74.5.
Frequency f = 55. Class interval i = 5.
Median = L + [(N/2 - cf) / f] × i
= 74.5 + [(50 - 22) / 55] × 5
= 74.5 + (28/11) = 74.5 + 2.54 = 77.04.
Ans: 77

12. Earth Geometry & 3D Shapes

(a) Earth Distance (Latitude):
Difference in longitude = 40° - 30° = 10°.
Latitude θ = 60°N.
Formula: Distance = (Difference in Longitude) × 111.6 × cos θ.
= 10 × 111.6 × cos 60°
= 1116 × 0.5
Ans: 558 km

(b) Rectangular Box:
Dimensions: 40cm, 30cm, 50cm.
(i) Length AG (Diagonal):
d = √(40² + 30² + 50²) = √(1600 + 900 + 2500) = √5000.
Ans: 50√2 cm
(ii) Total Surface Area:
SA = 2(lw + lh + wh) = 2(40×30 + 40×50 + 30×50)
SA = 2(1200 + 2000 + 1500) = 2(4700)
Ans: 9,400 cm²

13. Matrices

(a) Singular Matrix:
Determinant must be zero.
(-1)(3) - (11)(a) = 0
-3 - 11a = 0 → 11a = -3
Ans: a = -3/11

(b) Simultaneous Equations:
System: 5x + 3y = 9 and 10x + 7y = 11.
Matrix form AX = B. Determinant |A| = (5×7) - (3×10) = 5.
Inverse A^-1 = (1/5) × [7, -3; -10, 5].
Multiply A^-1 by B:
x = (1/5) [7(9) + (-3)(11)] = (1/5)(63 - 33) = 30/5 = 6.
y = (1/5) [-10(9) + 5(11)] = (1/5)(-90 + 55) = -35/5 = -7.
Ans: x = 6, y = -7

14. Functions and Linear Programming

(a) Inverse Functions:
If f(x) = x + 2, then f^-1(x) = x - 2.
If g(x) = 3x - 2, let y = 3x - 2 → 3x = y + 2 → x = (y+2)/3. So g^-1(x) = (x+2)/3.
Product = (x - 2) × [(x + 2)/3] = (x² - 4) / 3.
Ans: (x² - 4) / 3

(b) Linear Programming:
Let x = number of suits, y = number of gowns.
Constraints:
1. Cotton: 1x + 3y ≤ 150
2. Wool: 2x + 1y ≤ 90
3. Non-negativity: x ≥ 0, y ≥ 0.
Objective Function (Maximize Sales):
Z = 40,000x + 60,000y.

Disclaimer: These solutions are suggested for revision purposes.